Subsurface modeling for petroleum exploration or geological modeling often uses a layer cake velocity model consisting of layers of time horizons and the zones between them. These depict the velocity at different depths and thus the travel time of seismic waves, acoustic or vibrational, artificially generated to “see” the structures and features of the underground. Such a velocity model defined in time has an internal velocity field, for example, a velocity cube that represents a kind of 3-dimensional matrix corresponding to the subsurface earth volume, and used for converting (e.g., back and forth) between time and depth parameters when constructing a subsurface model. An average velocity cube, for example, enables domain-conversion of entire surfaces, not just individual data points rendered one at a time.
The boundaries or horizons of underground layers are approximated fairly well from the seismic reflection data. The approximated horizons, however, are not nearly as exact as the known depths of certain data points known as well markers. What is needed is a way to reconcile an entire velocity cube with the exactness of hard well marker data, so that the corrected velocity cube, in turn, can accurately model entire underground surfaces and volumes.